$C^{1,\alpha}$ isometric extensions

TL;DR

This paper develops a method to extend smooth isometric immersions to less regular $C^{1,eta}$ isometric immersions using convex integration, broadening the understanding of isometric embedding regularity.

Contribution

It introduces a convex integration technique to construct $C^{1,eta}$ isometric extensions from smooth immersions for the first time.

Findings

Successfully constructs $C^{1,eta}$ isometric extensions for $eta<rac{1}{n(n+1)+1}$

Demonstrates the applicability of convex integration to isometric extension problems

Provides new insights into the regularity thresholds for isometric immersions.

Abstract

In this paper we consider the Cauchy problem for isometric immersions. More precisely, given a smooth isometric immersion of a codimension one submanifold we construct $C^{1,\alpha}$ isometric extensions for any $\alpha<\frac{1}{n(n+1)+1}$ via the method of convex integration.